You would need a negative voltage (but not necessarily a completely bipolar power supply) to get the LM317 to go to zero. Dave fell madly in love with the Linear LT3080, which is a similar kind of regulator but it uses a current source to form it's voltage reference than a fixed 1.25V reference. It can get very very close to 0V. First thing that comes to mind to make a DIGITAL constant-current setting would be a digital potentiometer, but there might be better ways. I never got the impression that digital potentiometers were very good. . . often inaccurate, non-linear, not terribly robust.
Dave's power supply series is rather informative - check it out.
EDIT - If you want to try the LT3080, I have a fairly large number of them and I'd send a few to you at cost. I sold a few lots of 5 on ebay and through here a fair time ago because they were relatively hard to get your hands on. I just bought 100 direct from Linear and offloaded some of them to help folks obtain them. I still have plenty.